Taming Fat-Tailed (“Heavier-Tailed” with Potentially Infinite Variance) Noise in Federated Learning

Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track

Bibtex Paper Supplemental


Haibo Yang, Peiwen Qiu, Jia Liu


In recent years, federated learning (FL) has emerged as an important distributed machine learning paradigm to collaboratively learn a global model with multiple clients, while keeping data local and private. However, a key assumption in most existing works on FL algorithms' convergence analysis is that the noise in stochastic first-order information has a finite variance. Although this assumption covers all light-tailed (i.e., sub-exponential) and some heavy-tailed noise distributions (e.g., log-normal, Weibull, and some Pareto distributions), it fails for many fat-tailed noise distributions (i.e., ``heavier-tailed'' with potentially infinite variance) that have been empirically observed in the FL literature. To date, it remains unclear whether one can design convergent algorithms for FL systems that experience fat-tailed noise. This motivates us to fill this gap in this paper by proposing an algorithmic framework called $\mathsf{FAT}$-$\mathsf{Clipping}~$ (\ul{f}ederated \ul{a}veraging with \ul{t}wo-sided learning rates and \ul{clipping}), which contains two variants: $\mathsf{FAT}$-$\mathsf{Clipping}~$ per-round ($\mathsf{FAT}$-$\mathsf{Clipping}$-$\mathsf{PR}$) and $\mathsf{FAT}$-$\mathsf{Clipping}~$ per-iteration ($\mathsf{FAT}$-$\mathsf{Clipping}$-$\mathsf{PI}$). Specifically, for the largest $\alpha \in (1,2]$ such that the fat-tailed noise in FL still has a bounded $\alpha$-moment, we show that both variants achieve $\mathcal{O}((mT)^{\frac{2-\alpha}{\alpha}})$ and $\mathcal{O}((mT)^{\frac{1-\alpha}{3\alpha-2}})$ convergence rates in the strongly-convex and general non-convex settings, respectively, where $m$ and $T$ are the numbers of clients and communication rounds. Moreover, at the expense of more clipping operations compared to $\mathsf{FAT}$-$\mathsf{Clipping}$-$\mathsf{PR}$, $\mathsf{FAT}$-$\mathsf{Clipping}$-$\mathsf{PI}~$ further enjoys a linear speedup effect with respect to the number of local updates at each client and being lower-bound-matching (i.e., order-optimal). Collectively, our results advance the understanding of designing efficient algorithms for FL systems that exhibit fat-tailed first-order oracle information.